| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
27 Aug 2005 11:34:39 PM |
| Object: |
Quantum Complex Stochastics 2 |
From Osher Doctorow
Bassi (2003) generalizes non-Markovian sochatic Schrodinger equations
to any type of complex Gaussian noise and recovers the results of 8
papers from 1996 through 2002 in the process. These are flexible
formulations which cover important special cases of purely real and
purely imaginary noises or a suitable form of noise depending on the
particular problem studied.
Bassi's paper is only 4 pages long, but he points out that stochastic
Schrodinger equations have received considerable attention from the
1980s onward, although he only cites 1980s papers and most of the other
papers are from the 1990s onward (the majority from the mid-1990s).
He claims that a solution to the well known measurement problem of QM
was obtained by adding commuting operators coupled to stochastic noises
into the Schrodinger equation, driving the state vector into one of the
common eigenmanifols of these operators, which allows combining quantum
evolution and wavepacket reduction into a single dynamical equation.
Another advantage to stochastic Schrodinger equations has been in the
study of open quantum systems, defined as quantum systems interacting
with the surrounding environment. These are better than the standard
formalism of reduced density matrix which trace away all degrees of
freedom (dfs) of the environment, from the two viewpoints of not
reducing everything to aggregate ensembles and also by automatically
getting a positive definite operator describing the ensemble of
solutions of the stochastic Schrodinger equations which isn't
guaranteed by other approaches.
The stochastic Schrodinger equations used by Bassi turn out to have a
nice generalized Riccati Differential equation form, which will
hopefully be discussed later.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Complex Stochastics 2 |
28 Aug 2005 01:10:19 AM |
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From Osher Doctorow
For white noise, the stochastic Schrodinger equation has the form:
1) Dt |w(t)> = [-(i/h)H + f(L, z(t))]|w(t)>
with H the free Hamiltonian of the system being studied and f be a
function of operators L = (definition) {L1, L2, ..., Ln} and well as a
function of stochastic noise z(t) = (def.) {z1(),z2(t),..., zn(t)}.
Recently, as Bassi points out, this has been generalized to non-white
complex Gaussian noises which are descriptions of non-Markovian
evolutions, and the equation (1) generalizes to:
2) Dt|w(t)> = [-(i/h)H + L . z(t) - L^(dagger) . I ds a(t,s) .
delta/delta z(s)] |w(t)>
with z(t) complex 0-mean Gaussian noises with the dot being the product
between corresponding tensorial quantities such as L . z(t) = sum
Lizi(t) (sum over i) and the correlation functions between zi*(t)zj(z)
(the product enclosed in double << >> ) being aij(t,s) and between
zi(t)zj(s) being 0.
Equation (2) gives the correct predictions provided that the
probability distribution P(z(t)) associated to the stochastic process
of (2) is replaced by:
3) Q[z(t)] = P[z(t)]//w(t)//^2
which is an unusual breakthrough (3) in describing many types of
systems in many types of physical situations.
Bassi then studies more general stochastic noise.
Readers should look up Gaussian processes, which aren't the same as
Gaussian/normal probability distributions or probability density
functions. An upper division stochastic process course (usually a
second upper division probability course - that is, 3rd or 4th year) or
a textbook at the level of Port, Hoel, and Stone's Stochastic Processes
volume in the early 1970s (the authors' names might be permuted) would
explain that, or I'll try to do that (it isn't complicated). The words
"stochastic process" or "process" used with Gaussian or Poisson or
similar qualifiers isn't as general as the word "stochastic" used
without the qualifier "process", another example of complications of
syntax.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Complex Stochastics 2 |
28 Aug 2005 01:30:56 AM |
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From Osher Doctorow
More general stochastic Schrodinger equations are also interesting
because there are infinitely many different such equations leading to
the same equation for the statistical operator rho(t) and thereby
reproducing the same statistical behavior. For example, in the white
noise case, assuming for simplicity that operators L are self-adjoint,
equation (1) of last time becomes:
1) Dt|w(t)> = [-(i/h)H + L . z(t) - (1/2)L^2] |w(t)>
with correlation functions a(t,s) now being componentwise aij(t,s) =
delta_(ih) delta(t-s) with the Kronecker delta function. This isn't
norm-preserving for the norm of the state vector, the norm preserving
equation being for physical vectos |phi(t)>:
2) Dt|phi(t)> = [-(i/h)H + (L - <L> . (z(t) + <L>) - (1/2)(L^2 -
<L^2>)]|phi(t)>
with <L> = <phi(t)|L|phi(t)>, whih leads to the equation for the
statistical operator:
3) Dt rho(t) = -(i/h)[H . rho(t)] - (1/2)[L[L, rho(t)]]
which is of Lindblad type (see G. Lindblad, Commun. Math Phys. 48, 119
(1976).
The statistical operator rho(t) had been defined earlier in the white
noise scenario as the ensemble mean with regard to the noises z(t) of
the projection operators |w(t)><w(t)>:
4) rho(t) = <<|w(t)><w(t)|>>
with << >> denoting stochastic average. In that context, Dt(rho(t)) id
-(i/h)[H, rho(t)] + <<T[|w(t)><w(t)|]>> where the form of operator T
tepends on the operators L and noises z(t). For statevector collapse
models, T describes the average effect of wavepacket reductions, while
for open quantum systems it describes the effect of the environment on
the system. Note that the white noise case is Markovian, so the later
results generalize this.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Complex Stochastics 2 |
28 Aug 2005 01:39:44 AM |
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From Osher Doctorow
In the line immediately below (4) of last time, I wrote Dt(rho(t)) id.
I meant to write "is" instead of "id". This is not a Freudian slip,
but I don't know why exactly I made the typo (coincidence? Preceding
Dt? ).
Osher Doctorow
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